Finite Group Databasebeta

← #22600039

Subgroups — #22600039

Order 9088.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D71C212yesmaximal, complemented
2#17986664C211yesmaximal, characteristic, complemented
3D4×D284C211yesmaximal, characteristic
4#17986373C211yesmaximal, characteristic, complemented
5#17986231C212yesmaximal, complemented
6#17986200C211yesmaximal, characteristic, complemented
7#17986148C212yesmaximal, complemented
8#17986139C212yesmaximal, complemented
9C4×#11765853C211yesmaximal, characteristic, complemented
10#17986064C211yesmaximal, characteristic, complemented
11#461×C71C211yesmaximal, characteristic, complemented
12C2×Q8×D71C2212yescomplemented
13C22×D284—21no—
14C2×D4×D71C2212yes—
15C2×#11765854C2212yescomplemented
16C2×#11765853C2211yescharacteristic
17C2×#11765852C2212yescomplemented
18C2×#11765852C2212yescomplemented
19SD16×D71—28no—
20C2×C8×D71C2212yescomplemented
21#14556127—21no—
22#14556125C2211yescharacteristic
23#14556106—22no—
24#14556104C2212yescomplemented
25#14556103C2212yes—
26#14556102—21no—
27#14556098—22no—
28C4×Dic142C2211yescharacteristic, complemented
29C4×D284C2211yescharacteristic
30#14556090—21no—
31#14556059C2211yescharacteristic, complemented
32#14556051C2211yescharacteristic, complemented
33#14556050C2212yescomplemented
34#14556049C2211yescharacteristic
35(C2×D71)⋊C8C2212yescomplemented
36C4×C71⋊C8C2211yescharacteristic, complemented
37C2×SD16×C71C2212yes—
38C4⋊Q8×C71C2211yescharacteristic, complemented
39C4×D4×C71C2211yescharacteristic
40C4⋊C8×C71C2211yescharacteristic, complemented
41#103×C71C2212yescomplemented
42C23×D71—22no—
43C2×Dic142—22no—
44C2×Dic142C2311yescharacteristic
45Q8×D71—24no—
46Q8×D71—42no—
47C2×D284—22no—
48C2×D284—21no—
49C2×D284—21no—
50C2×D284—44no—
51C2×D284D411yescharacteristic
52C2×D284C2312yes—
53C2×C22⋊D71—22no—
54D4×D71—24no—
55D4×D71—42no—
56C2×C4×D71—22no—
57C2×C4×D71C2312yes—
58#11765854—24no—
59#11765853D414yescomplemented
60#11765852—24no—
61#11765852—42no—
62C2×C71⋊C8C2311yescharacteristic
63C2×C71⋊C8C2311yescharacteristic
64C8×D71—24no—
65C4⋊Dic71D411yescharacteristic
66C4⋊Dic71C2311yescharacteristic
67Dic71⋊C4—21no—
68Dic71⋊C4—22no—
69(C2×D71)⋊C4—22no—
70(C2×D71)⋊C4—22no—
71(C2×D71)⋊C4D412yes—
72C4×Dic71—21no—
73C2×Q8×C71C2312yes—
74C2×D4×C71C2311yescharacteristic
75C22×C4×C71—21no—
76SD16×C71—24no—
77C2×C8×C71C2312yes—
78C4⋊C4×C71—21no—
79C4⋊C4×C71—21no—
80C4⋊C4×C71C2311yescharacteristic
81C22⋊C4×C71—21no—
82C42×C71C2311yescharacteristic
83C23×C71—21no—
84C22×D71—22no—
85C22×D71—44no—
86C22×D71—44no—
87C22×D71—82no—
88C22×D71C2×D412yes—
89Q8×C71—22no—
90Q8×C71—22no—
91D4×C71—21no—
92D4×C71C2×D412yes—
93C2×C4×C71—21no—
94C2×C4×C71—22no—
95C2×C4×C71—42no—
96C2×C4×C71C2×D411yescharacteristic
97C2×C4×C71C2×D411yescharacteristic
98C2×C4×C71C2411yescharacteristic
99C22⋊D71—44no—
100C2×Dic71—21no—
101C2×Dic71—21no—
102C2×Dic71C2×D412yes—
103D284—22no—
104D284—22no—
105D284—44no—
106D284—42no—
107D284—81no—
108D284SD1614yescomplemented
109C4×D71—22no—
110C4×D71—22no—
111C4×D71—44no—
112Dic142—21no—
113Dic142—44no—
114Dic142C2×D412yescomplemented
115C8×C71—22no—
116C71⋊C8—21no—
117C71⋊C8C2×D412yescomplemented
118C22×C71—22no—
119C22×C71—41no—
120C22×C71—41no—
121C22×C71C22×D411yescharacteristic
122C2×D71—22no—
123C2×D71—22no—
124C2×D71—42no—
125C2×D71—42no—
126C2×D71—42no—
127C2×D71—88no—
128C2×D71C2×SD1614yescomplemented
129C4×C71—21no—
130C4×C71—42no—
131C4×C71—41no—
132C4×C71C2×SD1612yescomplemented
133C4×C71C22×D411yescharacteristic
134C4×C71H(5,𝔽2)11yescharacteristic
135Dic71—22no—
136Dic71—22no—
137Dic71—41no—
138C2×C71—42no—
139C2×C71D4211yescharacteristic
140C2×C71C22×SD1611yescharacteristic
141C2×C71#57711yescharacteristic
142D71—24no—
143D71—82no—
144D4×SD16—711nomaximal, Sylow
145C71D4×SD1611yescharacteristic, complemented, Sylow
146C22×SD16—712no—
147D4×Q8—711no—
148D42—711no—
149#492—711no—
150#465—712no—
151#461D7111yescharacteristic, complemented
152#460—711no—
153#450—712no—
154#448—712no—
155C4×SD16—711no—
156C8×D4—711no—
157C22×Q8—712no—
158C22×D4—712no—
159C22×D4—1421no—
160C2×SD16—712no—
161C2×SD16—711no—
162C2×SD16—712no—
163C2×SD16—712no—
164C2×SD16—1428no—
165C2×SD16C2×D7112yes—
166C22×C8—712no—
167C4⋊Q8—1421no—
168C4⋊Q8C2×D7111yescharacteristic, complemented
169#127—711no—
170C22⋊Q8—712no—
171C22⋊Q8—1422no—
172#121—712no—
173#121—1421no—
174#120—1422no—
175C4×Q8—711no—
176C4×D4—711no—
177C4×D4—1421no—
178C4×D4C2×D7111yescharacteristic
179#106—711no—
180C4⋊C8C2×D7111yescharacteristic, complemented
181#103—711no—
182#103C2×D7112yescomplemented
183#102—711no—
184#102—712no—
185C22⋊C8—712no—
186C4×C8—711no—
187C24—1422no—
188C2×Q8—2842no—
189C2×Q8—711no—
190C2×Q8—1422no—
191C2×Q8—1424no—
192C2×Q8C22×D7112yes—
193C2×D4—2844no—
194C2×D4—2842no—
195C2×D4—711no—
196C2×D4—712no—
197C2×D4—1422no—
198C2×D4—1422no—
199C2×D4—1424no—
200C2×D4—1421no—
201C2×D4—1421no—
202C2×D4C22×D7111yescharacteristic
203C22×C4—21no—
204C22×C4—712no—
205C22×C4—1422no—
206SD16—24no—
207SD16—2842no—
208SD16—714no—
209SD16—1424no—
210SD16—1424no—
211C2×C8—711no—
212C2×C8—711no—
213C2×C8—1424no—
214C2×C8C22×D7112yes—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—711no—
218C4⋊C4—711no—
219C4⋊C4—1422no—
220C4⋊C4—1421no—
221C4⋊C4C22×D7111yescharacteristic
222C22⋊C4—21no—
223C22⋊C4—712no—
224C22⋊C4—1422no—
225C22⋊C4—1422no—
226C42—1421no—
227C42C22×D7111yescharacteristic
228C23—21no—
229C23—2844no—
230C23—2844no—
231C23—5682no—
232C23—712no—
233C23—1422no—
234Q8—22no—
235Q8—22no—
236Q8—2844no—
237Q8—712no—
238Q8—1421no—
239D4—21no—
240D4—2844no—
241D4—2844no—
242D4—2842no—
243D4—5681no—
244D4—714no—
245D4—1422no—
246D4—1422no—
247D4D4×D7112yes—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—42no—
251C2×C4—2844no—
252C2×C4—712no—
253C2×C4—1421no—
254C2×C4—1421no—
255C2×C4—1422no—
256C2×C4—1422no—
257C2×C4D4×D7111yescharacteristic
258C2×C4D4×D7111yescharacteristic
259C2×C4C23×D7111yescharacteristic
260C8—22no—
261C8—712no—
262C8—1421no—
263C22—22no—
264C22—41no—
265C22—41no—
266C22—2842no—
267C22—2842no—
268C22—2842no—
269C22—5688no—
270C22—714no—
271C22—1422no—
272C22—1422no—
273C22C2×D4×D7111yescharacteristic
274C4—21no—
275C4—42no—
276C4—41no—
277C4—2841no—
278C4—1422no—
279C4—1422no—
280C4SD16×D7112yescomplemented
281C4C2×D4×D7111yescharacteristic
282C4#1455617411yescharacteristic
283C2—42no—
284C2—5682no—
285C2—1424no—
286C2#1798663311yescharacteristic
287C2C2×SD16×D7111yescharacteristic
288C2#1798680811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#461×C71C2
socleC22×C71C22×D4
hypercenter#461D71
hypocenterC71D4×SD16
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×SD16×D71C2semidirect
2. #17986664C2semidirect
4. #17986373C2semidirect
5. #17986231C2semidirect
6. #17986200C2semidirect
7. #17986148C2semidirect
8. #17986139C2semidirect
9. C4×#11765853C2semidirect
10. #17986064C2semidirect
11. #461×C71C2semidirect
12. C2×Q8×D71C22semidirect
15. C2×#11765854C22semidirect
17. C2×#11765852C22semidirect
18. C2×#11765852C22semidirect
20. C2×C8×D71C22semidirect
24. #14556104C22semidirect
28. C4×Dic142C22semidirect
31. #14556059C22semidirect
32. #14556051C22semidirect
33. #14556050C22semidirect
35. (C2×D71)⋊C8C22semidirect
36. C4×C71⋊C8C22semidirect
38. C4⋊Q8×C71C22semidirect
40. C4⋊C8×C71C22semidirect
41. #103×C71C22semidirect
59. #11765853D4semidirect
108. D284SD16semidirect
114. Dic142C2×D4semidirect
117. C71⋊C8C2×D4semidirect
128. C2×D71C2×SD16semidirect
132. C4×C71C2×SD16semidirect
145. C71D4×SD16semidirect
151. #461D71semidirect
168. C4⋊Q8C2×D71semidirect
180. C4⋊C8C2×D71semidirect
182. #103C2×D71semidirect
280. C4SD16×D71semidirect